Two Best-Prep Math Concepts
Many problems on your exam will require performing more than one mathematical operation. Depending on the order you choose to evaluate these operations, your answer will vary. Since there is only one correct answer, it is crucial that you follow the ìGirl Scouts Simple Order of Operations.î When operations have the same rank, ìyield to the left,î that is, do whatís on the left first. The Girl Scout Simple Order is:
1.†† The GIRL SCOUT ó Grouping symbols are evaluated first. Grouping symbols may come in pairs. They may be parenthesis ( ), brackets [ ], or braces { }. Groups may also be separated by a fraction bar, or a radical sign.† If there is more than one set of groups, evaluate the innermost first. Think of a Magician who is inside a paper bag and put into a suitcase. He needs to get out of the bag first and then he can get out of the suitcase.
2.†† Who GAVE POWER ó Powers of exponents are ranked second.
3.†† To MY DEAR ó Multiplication and division are both ranked third.
4. AUNT SALLY ó Addition and subtraction are both ranked last. We save the easiest operations for last.
And got a good EVALUATION.
1. GS:5 + 23 ˜ 4 + (-1)2 x 2 - 32
2. Powers: 5 + 8 ˜ 4 + 1 x 2 - 9
3: MD† 5 + 2+ 2- 9
4. AS0
Try These:
2. If for all real numbers a and b, a $ b† =† a2 - 2ab + b2 then 3 $ 4† =
a) 1b) -1c) 49d) 0e) - 7
Hint: Substitute 3 for a, and 4 for b, then evaluate away!
[1]SOLVING EQUATIONS AND
INEQUALITIES ñMake x happy!
An equation is a sentence with an equals sign. It can be thought of as a balanced scale.
The object of solving an equation is to make a variable HAPPY. A HAPPY variable is all alone one side of an equals sign. Using the following Commandments of mathematics, it is easy to obtain a HAPPY variable.
Math Commandments:
I. Undo what has been done by doing the opposite operation in the opposite order.
II. Do unto one side of an equation as you would do unto the other.
III. Distribute fairly: Multiplication is distributive over addition† or
†a (b + c)† =† ab + ac
e.g.††† 8( 101) = 8 (100 + 1) = 800 + 8 = 808
SOLVE: 3N + 8† =† 20
SOLUTION: 3N + 8 = 20
In the expression above, 3N + 8 finds N having been multiplied by 3 then 8 being added on. Remember the order of operations? To UNDO this we can first subtract 8 (or add 8), then divide by 3 (or multiply by 1/3) -- as they say..Last hired, first fired.
3N + 8 = 20
†- 8-8
subtract 8 from both sides obtaining
3N =12
3 3
dividing both sides by 3 gives us
N =4
Check it out: Go back and substitute 4 for N and see if you get the left side to equal the right side.
A statement with a < or> symbol in it is called an inequality. Inequalities may be solved using the same 2 rules as above.
if 3N + 8 < 20then N < 4
The only difference comes when you need to multiply both sides of an inequality by a negative number. That is, a negative number is squishing or squashing poor x. When this happens, a change in direction is needed.
EXAMPLE:ifñ3N < 12then N > 4
Now it is your turn: [A solution follows, but why not try it on your own first?]
A Solution:
(P - 5) + 2/5 (P +5)=4
Generally speaking, it is easier to get rid of fractions first when solving equations and making the variable happy. This can be done using commandments I and III.
First We will multiply both sides of the equation by 5.
This gives us
(1) 5(P - 5) + 2 (P + 5) +=20
[Notice that when we distribute 5 to the term: 2/5 (P +5), it only goes to the first factor (2/5) giving us:
2/5 (2) = 2.It does not effect the second factor, (P + 5).
Next, Letís apply the distributive property Commandment III -- again.
(2) 5P - 25 + 2P + 10= 20 Collecting like terms, we get:
Adding 15 to both sides† -- Commandment I -- gives us
Dividing both sides by 7 -- Commandment I againógives us:
Check this out by going back to the original equation and evaluating the left side
when P = 5!